3-D vessel tree surface reconstruction

ABSTRACT

Reconstructing 3-D vessel geometry of a vessel includes: receiving a plurality of 2-D rotational X-ray images of the vessel; extracting vessel centerline points for normal cross sections of each of the plurality of 2-D images; establishing a correspondence of the centerline points from a registration of the centerline points with a computed tomography (CT) 3-D centerline, the registration being an affine or deformable transformation; constructing a 3-D centerline vessel tree skeleton of the vessel from the centerline points of the 2-D images; constructing an initial 3-D vessel surface having a uniform radius normal to the 3-D centerline vessel tree skeleton; defining sample points based sampling on median radii to the 3-D centerline vessel tree skeleton of the initial 3-D vessel surface; and constructing a target 3-D vessel surface by deforming the initial vessel surface using the sample points to provide a reconstructed 3-D vessel geometry of the vessel.

BACKGROUND

The present disclosure relates generally to a three-dimensional (3-D) vessel tree surface reconstruction method, particularly to a 3-D coronary artery tree surface reconstruction method, and more particularly to a 3-D coronary artery tree surface reconstruction method from a limited number of two-dimensional (2-D) rotational X-ray angiography images.

In percutaneous coronary intervention (PCI) procedures, physicians evaluate and identify coronary artery lesion (stenosis), and prepare catheterization utilizing X-Ray coronary angiographic images. These images are 2-D projection images of a complex coronary artery tree acquired by X-Ray machines called C-Arms either from bi-plane or from mono-plane detector arrangements. 2-D projections cause vessel occlusion, crossing, and foreshortening. To better understand vessel geometry, multiple views with different angles may be acquired. In addition, 2-D projection image based quantitative coronary analysis (QCA) may determine lesion length and stent size during PCI. However, there are two major limitations of 2-D QCA: foreshortening and out-of-plane magnification errors.

3-D reconstruction of the vessel surface may be used to avoid the limitations of 2-D QCA. In some 3-D tomographic reconstructions of coronary arteries, motion artifact is minimized through a pre-computed 4-D motion field. The 4-D motion field is computed from 3-D coronary artery centerline reconstruction and a 4-D parametric motion model fitting. However, this tomographic-based approach is computationally expensive. In addition, using a limited number of images can cause a blur and low resolution reconstruction. In another approach, 3-D symbolic reconstruction uses two projections and multiple projections. A 3-D vessel skeleton is reconstructed, and then an elliptical model representing a vessel cross-section is fit using 2-D measurement (e.g. segmentation) or estimating vessel radii from 2-D measurement. Different centerline reconstruction approaches are used without computationally expensive surface reconstruction. Elliptical or circular models are symbolic and are not accurate due to lumen deformation and lesion. 2-D vessel lumen segmentation on the projection image is a challenging task due to vessel occlusion and crossing, so may require many iterations of user intervention.

U.S. Published Application No. 2017-0018116 discloses automatic generation of a 3-D vessel geometry for performing 3-D QCA in PCI procedures. Generation of the 3-D vessel geometry without full surface reconstruction may avoid the limitations from 2-D projection images. 3-D QCA allows quantitative determination of vessel lumen, grade of stenosis, and non-invasive calculation of fractional flow reserve (FFR). Rigid cardiac motion in ECG synchronized frames is assumed. The surface fit may be less accurate than desired in some cases.

SUMMARY

By way of introduction, the preferred embodiments described below include methods, non-transitory computer readable media, and systems for determining a 3-D vessel surface or geometry for 3-D QCA. Using 2-D projections to reconstruct the surface, non-rigid-based registration builds point-to-point correspondence. A hierarchical solution using variable sample points based on an initial surface reconstruction improves surface reconstruction accuracy.

In a first aspect, a method is provided for reconstructing a 3-D vessel geometry of a vessel. The method includes receiving a plurality of 2-D rotational X-ray images of the vessel; extracting vessel centerline points for normal cross sections of each of the plurality of 2-D images; establishing a correspondence of the centerline points from a registration of the centerline points with a computed tomography (CT) 3-D centerline or a rotational angiography (e.g., DynaCT) 3-D centerline, the registration being an affine or deformable transformation; constructing a 3-D centerline vessel tree skeleton of the vessel from the centerline points of the 2-D images; constructing an initial 3-D vessel surface having a uniform radius normal to the 3-D centerline vessel tree skeleton; defining sample points based sampling on median radii to the 3-D centerline vessel tree skeleton of the initial 3-D vessel surface; and constructing a target 3-D vessel surface by deforming the initial vessel surface using the sample points to provide a reconstructed 3-D vessel geometry of the vessel.

A second aspect includes a computer program storage medium comprising a non-transitory computer readable medium having program code executable by a processing circuit for implementing the above noted method.

A third aspect includes an apparatus for reconstructing 3-D vessel geometry of a vessel, comprising: a computer controlled machine comprising a processing circuit; and computer readable executable instructions, which upon loading into the processing circuit causes the machine to be responsive to the executable instructions to facilitate implementation of the above noted method.

The above features and advantages and other features and advantages of the invention are readily apparent from the following detailed description of the invention when taken in connection with the accompanying drawings.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

Referring to the exemplary non-limiting drawings wherein like elements are numbered alike in the accompanying Figures:

FIG. 1 depicts a model using a 2-D vessel cross section to illustrate vessel surface deformation in accordance with an embodiment;

FIG. 2(a) depicts two distinct vessels crossing each other due to camera projection for use in accordance with an embodiment;

FIG. 2 (b) depicts the normal directions for two overlapping (very close) points that belong to different vessel branches for use in accordance with an embodiment;

FIG. 3 depicts a bipartite graph corresponding to a linear assignment process in accordance with an embodiment;

FIG. 4 depicts a point cloud construction around the centerline of a vessel tree skeleton in accordance with an embodiment;

FIG. 5(a) depicts an initial vessel surface (synthetic vessel mesh surface, e.g. a tubular structure with uniform radius) in accordance with an embodiment;

FIG. 5(b) depicts a mesh zoom-in of the mesh of FIG. 5(a) in accordance with an embodiment;

FIG. 5(c) depicts vessel point sampling in 3-D space as performed along a searching profile (e.g. vertex normal direction) for a vertex V_(i) in accordance with an embodiment;

FIG. 5 (d) depicts a graph structure that is constructed based on sampling nodes in accordance with an embodiment;

FIGS. 6(a), 6(d), 6(g) and 6(j), the first column, depicts original 2-D rotational X-ray angiography images from four different patients for use in accordance with and embodiment;

FIGS. 6(b), 6(e), 6(h) and 6(k), the second column, depicts reconstructed vessel surfaces from the same view as the acquired original images in accordance with an embodiment;

FIGS. 6(c), 6(f), 6(i) and 6(l), the third column, depicts the projected 2-D segmentation overlays from reconstructed surfaces in accordance with an embodiment;

FIG. 7 depicts a method and system in accordance with an embodiment;

FIGS. 8(a), 8(b) and 8(c), depict an apparatus for capturing 2-D rotational X-ray angiography images in accordance with an embodiment;

FIGS. 9(a), 9(c) and 9(e) depict reconstructed vessel surfaces from the same view as the acquired original images in accordance with an embodiment using affine transformation and a hierarchal surface reconstruction; and

FIGS. 9(b), 9(d) and 9(f) depict the projected 2-D segmentation overlays from reconstructed surfaces from FIGS. 9(a), 9(c), and 9(e), respectively, in accordance with an embodiment.

DETAILED DESCRIPTION

Although the following detailed description contains many specifics for the purposes of illustration, anyone of ordinary skill in the art will appreciate that many variations and alterations to the following details are within the scope of the claims. Accordingly, the following example embodiments are set forth without any loss of generality to, and without imposing limitations upon, the claimed invention.

To improve accuracy and computational efficiency of the embodiments of U.S. Published Application No. 2017-0018116, registration with affine or thin-plate spline transformation is used. A hierarchal approach to 3-D vessel surface reconstruction adds an additional act with a less strict smoothness constraint and more flexibility in adapting to x-ray images and vesselness is alternatively or additionally provided. The added surface reconstruction act may increase detail in the 3-D vessel surface.

Below, the generalized pipeline for 3-D vessel surface reconstruction from rotational coronary x-ray angiography images is provided. The improvements to this pipeline (i.e., different transformations for registration of a CT 3-D centerline with centerlines from 2-D projection images and an added hierarchal reconstruction step) are also discussed (see acts 706, 711, and 713 for the improvements).

The pipeline without the improvements utilizes ECG signals to extract 2-D frames of the same cardiac phase in the rotational sequence. 2-D scenes at a cardiac phase are compared to a CT 3-D centerline, such as based on DynaCT. To reconstruct the surface based on this comparison, the image processor performs segmentation, registration, sampling in 3-D, and constructing the surface through a smoothness constrained graph search. The resulting vessel surface is satisfactory in its smoothness and spatial location. Registration speed and surface details could be improved. The registration is improved using affine or warping transformation, and the surface details are improved by adding a surface reconstruction act that uses center point specific radii in the hierarchy of surface reconstruction.

For the pipeline, an embodiment includes a method to reconstruct a 3-D vessel tree surface, coronary artery for example, from a limited number of 2-D rotational X-ray angiography images. First, a limited number of views with a same cardiac phase are identified from the X-ray angiography images using Electrocardiography (ECG) signals (or ECG gated acquisition signals). Second, vessel centerlines of those 2-D images are extracted utilizing a known technique of model-guided extraction of coronary vessel structures in 2-D X-ray angiograms. The correspondence of centerline points are identified using a prior 3-D vessel shape segmented from DynaCT reconstruction, using shape properties such as vessel tangents and normals, and solving an assignment problem. The best match between a prior 3-D model constrains the assignment problem. The best match is improved with an affine or thin-plate spline transformation. In one embodiment, the prior segmented shape can be obtained from a DynaCT volume. For the generation of a DynaCT volume, known approaches such as interventional 4D motion estimation and reconstruction of cardiac vasculature without motion periodicity assumption, or ECG-gated interventional cardiac reconstruction for non-periodic motion, and considering tomographic 3-D/4-D reconstruction may be utilized. In an embodiment, only one data set is needed for point correspondence and 3-D coronary tree surface reconstruction. Third, a 3-D centerline (vessel tree skeleton) is reconstructed from 2-D points using a bundle adjustment based approach. Fourth, the vessel surface is reconstructed by finding an optimal surface for the vessel tree skeleton. The sought surface is optimized based on cost functions derived from 2-D images and constrained by the vessel skeleton. To improve the surface detail, sampling points after an initial iteration or surface is created are defined based on clustering nodes of the initial surface mesh to the closest 3-D centerline vessel tree skeleton points, and the medians distance from the surface points to the centerline skeleton point for each skeleton point define a radius of a circle of sample points for that skeleton point. The reconstruction proceeds using these variable radii sample points.

Features of an embodiment of the method disclosed herein include: (1) a fully automated vessel surface reconstruction method (overall pipeline); (2) an alignment estimation between prior models and multiple segmented 2-D structures (a benefit of an embodiment disclosed herein is that multiple landmarks are identified and detected in many views, compared to other techniques that consider only one landmark, which contemplates a significant improvement in accuracy); (3) a centerline reconstruction algorithm is proposed (a reconstructed 3-D centerline is expected to have higher precision and accuracy for lumen surface reconstruction than a centerline previously derived from DynaCT even ECG-gating from the same cardiac phase being utilized, which contemplates more accurate projections resulting from a centerline reconstruction process); (4) an assignment problem constrained by using model alignment that considers vessel tangents and normals during the alignment phase; (5) an embodiment includes a best alignment based between multiple 2-D projections and a 3-D model based on costs derived from vessel tangents and normals and from affine or thin-plate spline transformation; (6) an embodiment includes a best assignment by utilizing a prior 3-D model as a constraint; (7) while the existing mainstream of vessel surface reconstruction methods are from 2-D to 3-D, an embodiment includes a reconstructed vessel surface in 3-D space driven by a data term that is derived from 2-D information via back-projection (existing methods rely on 2-D vessel segmentation based on which symbolic surface is constructed, which require performing 2-D vessel lumen segmentation or finding 2-D vessel boundary correspondence, where both problems are non-trivial due to projection limitations such as vessel occlusion and branch crossing); (8) an embodiment includes 3-D vessel surface reconstruction through solving an optimal graph search problem (the missed data term (cost function) due to a lack of camera views is regularized by a surface smoothness, which contemplates achieving a global optimal solution under a certain graph construction) where an added step with less strict smoothness constraint and more flexibility allows for more surface detail; (9) methods to find a cost function are presented; and, (10) with utilization of an embodiment of 3-D surface reconstruction as disclosed herein, 2-D vessel segmentation can be generated using back-projection.

Reference is now made to FIG. 1, which illustrates a model 100 in accordance with an embodiment using a 2-D vessel cross section (normal to a centerline of the vessel) to illustrate vessel surface deformation. The solid-line curve depicts a target vessel surface. The dotted-line curve 102 depicts an initial vessel boarder having a centerline point 104. An initial vessel surface (synthetic vessel surface, e.g. a tubular structure with uniform radius), also referred to by reference numeral 102, is constructed around the centerline 104 and evolves towards a true (target) vessel surface 106. In this way, the surface reconstruction problem (finding the true vessel surface 106) is transformed into finding an optimal vessel surface problem (establishing the synthetic initial vessel surface 102). The arrow 108 depicts a deformation direction of the initial vessel surface point of 102 that is driven by image information collected from all corresponding 2-D projection views. At the same time, surface smoothness is enforced to make sure a feasible surface results. As such, an embodiment of the method does not need to compute 2-D vessel segmentation or find sufficient corresponding 2-D vessel boundary points. The enforcement of the smoothness may be relaxed for one or more iteration in solving for the surface in order to provide more surface detail.

To be able to construct the 3-D vessel centerline 104, point correspondences need to be established between the segmented vessel points across the 2-D images. In an embodiment of the method, the vessel topology is extracted in the form of ordered point sets, Si, from each angiographic image, where i is i^(th) image. The method assumes that each point set Si is the projection of a transformed 3-D centerline model M according to the following: S _(i) =P _(i) T(M,θ ₁),  Equa. 1 where P_(i) is the camera projection operator corresponding to image i, and T(M, θ_(i)) is a defined transformation model. Given M and S_(i), the method employs Gaussian mixture models (GMM) to find transformations T(M, θ_(i)) that minimize the following cost function: J(S _(i) ,M,θ _(i))=∫(GMM(S _(i))−GMM(P _(i) T(M,θ _(i))))^(p) dx,  Equa. 2 where the energy function J(Si,M, θ_(i)) measures the distance of two Gaussian mixture models (GMM) where p∈[0, 2]. In some embodiments, when p=2, this cost function corresponds to the l₂-norm. In such embodiments, a closed-form solution can be utilized. In some embodiments, when p=1, the cost function corresponds to l₁-norm, corresponding with KL(Kullback-Leibler)-divergence of the two GMMs. Such embodiments can be more forgiving to outliers existing in either sets. The GMMs are obtained by treating each point as a separate Gaussian component with its mean being the spatial location of the point along with a user defined spherical covariance matrix.

The above framework estimates separate transformations for each of the angiographic images. However, when the 3-D model is not accurate, it introduces errors that propagate to point correspondences. In such cases, a more robust way would be to estimate a single transformation that aligns all of the angiographic images to an averaged motion model and hence, the minimization of the following energy function is utilized:

$\begin{matrix} {{\sum\limits_{i}^{N}{w_{i}{J\left( {S_{i},M,\theta} \right)}}},} & {{Equa}.\mspace{14mu} 3} \end{matrix}$ where w_(i)∈[0, 1], Σw_(i)=1, is the weight associated with image i. Images are analyzed to extract these weights.

Loss of depth due to camera projection can cause distinct vessels to fall on top of each other. One such example is depicted in FIG. 2(a), which depicts two distinct vessels 202, 204 crossing each other due to camera projection as shown in the two dashed boxes 206, 208. This phenomenon can cause the registration algorithm to get stuck in a poor local minima. Since the segmented point sets have structure, topological information can be used to discriminate between overlapping vessel structures. FIG. 2 (b) depicts the normal directions 210, 212 for the two overlapping (very close) vessels 202, 204 associated with box 208 in FIG. 2(a), represented by projected intersection point 214 in FIG. 2(b). Although these overlapping vessels share a similar projected location (projected intersection point 214), their normal directions 210, 212 are very different. This information is utilized in the registration framework by modifying each GMM component to include its normal vector in its mean. Note that this does not change the distance formulation given in Equa. 2. In this embodiment, the overlapping point location x_(i,j) (projected intersection point 214 for example) is augmented to include the unit normal, x_(i^,j) at the point. In some other embodiments, the point location can be augmented image derived features such as SURF (Speeded Up Robust Features), SIFT (Scale-Invariant Feature Transform) and FFT (Fast Fourier Transform), for example, at the respective local neighborhood.

Once the registration step is completed, point correspondences are established between the vessel points across the images. These correspondences are solved by finding the matching vessel point for every projected model point in each image. For each image and the model set, this matching is solved via a linear assignment process. As shown in FIG. 3, a first set of vertices 302 represents transformed, projected model points of the vessel centerline points (see 104 in FIG. 1 for example), and a second set of vertices 304 represents the associated segmentation points of a prior 3-D vessel shape of the vessel segmented from a DynaCT volume and DynaCT reconstruction. The vertices 302, 304 are represented as a bipartite graph 300 that corresponds to the linear assignment process. The weights of these edges, c_(ij), can be computed using the distance of spatial locations of the vertices. Furthermore, it can be assumed that if the two vertices are very far apart, they should not be matching. To incorporate this assumption, a distance threshold, d_(th), is used as follows: c _(i,j) ={d _(i,j) if d _(i,j) <dth; ∞ if d _(i,j—) d _(th)}.  Equa. 4

Such a thresholding operation makes certain matchings infeasible, and therefore provides robustness in the case of missing vessel segmentations.

Regarding centerline reconstruction, ECG gating helps to compensate cardiac motion, however, it should be noted that the respiratory (breathing) motion also needs to be compensated for to make sure that projection views are coherent with the 3-D reconstruction. The breathing motion can be approximated as a 3-D translation in accordance with known methods. An embodiment of the method compensates this motion by adapting camera projection parameters employing a known bundle adjustment based method. It should also be noted that the remaining rigid transformation due to non-perfect ECG gating can also be compensated in this process. An embodiment of the method iteratively reconstructs the 3-D vessel center-line from corresponding 2-D centerline points and performs bundle adjustment. 3-D centerline and refined camera parameters (3-D to 2-D projection matrix) are computed based on the following equation:

$\begin{matrix} {{\min\limits_{a_{j},b_{i}}{\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{m}{k^{i,j}{d\left( {{Q\left( {a_{j},b_{i}} \right)},x^{i,j}} \right)}}}}},} & {{Equa}.\mspace{14mu} 5} \end{matrix}$ where: n is the number of 3-D points that are seen in m views; camera j is parameterized by a vector a_(j); b_(i) is the i^(th) centerline point; Q(a_(j), b_(i)) is the projection of 3-D point b_(i) on image j; x^(i,j) is the i^(th) 2-D centerline point measurement on image j; d(,) is a Euclidean distance between the associated points defined within the parentheses; and, k^(i,j)∈[0, 1] is a confidence measurement of x^(i,j).

Reference is now made to FIGS. 4, 5(a) and 5(b) regarding surface reconstruction. Around the resulting 3-D centerline 402 (see also centerline point 104 in FIG. 1), an initial mesh surface is computed (best seen with reference to FIGS. 5(a) and 5(b) discussed further below). A point cloud construction 404 is generated around the centerline 402 as depicted in FIG. 4. The dotted-line circles (also referred to by reference numeral 404) depict sampling points around an associated centerline point 104 (see FIG. 1 for example) of centerline 402. It should be noted that these circles 404 are perpendicular to an associated centerline tangential direction 210, 212 (see FIG. 2(b) for example). Sampling points are normally distributed along a normal direction of the corresponding centerline point with radius r. In one embodiment, a Poisson surface reconstruction method is utilized to compute the mesh from the point cloud. An example of an initial mesh surface 502, 504 is depicted in FIGS. 5(a) and 5(b).

For each vertex on the initial mesh surface 504, a vector of points is created and contains the final sought surface vertex V. The sampling 506 is performed in the 3-D space along a searching profile for each initial vertex. Note that the searching profile intersects the initial surface covering both inside and outside initial surface. In an embodiment, the searching profile is along the vertex normal direction. The sampling method is illustrated in FIG. 5(c). In an embodiment, sampling distance can be uniform. In an embodiment, the sample distance linearly decreases with the radius. In the 3-D segmentation domain, a deformable shape model is often utilized to segment an object with the known shape. One example can be found in a U.S. patent publication no. 2002/0136440 (hereinafter referred to as the '440 publication), where a 3-D vessel shape is segmented using a deformable shape model in 3-D angiography image data. An embodiment of the method disclosed herein can also utilize a deformable shape model. However, a fundamental difference between an embodiment disclosed herein and the '440 publication is that the 3-D segmentation solution provided herein produces a surface reconstruction from a limited number of 2-D projection images, but not in a volumetric image. In an embodiment, the surface deformation is driven by 2-D information through 3-D to 2-D projection. Due to the lack of views, not all projected surface points are aligned with a 2-D vessel border, but instead are inside the vessel. In contrast to 3-D segmentation, an embodiment also addresses additional challenges like vessel occlusion and crossing problem.

Assume that each sampling point has a cost function w(v_(i), S(v_(i))) indicating that the sampling point is unlikely on the sought surface, where v_(i) is the surface vertex and S(v_(i)) is any sampling point along a searching profile for v₁. We denote T={S(v_(i))|i=1 . . . L} as an arbitrary (initial) vessel surface, where L is the number of vertices. The method solves the surface reconstruction by finding a minimum-surface cost subject to smoothness constraints |S(v_(i))−S(v_(j))|≤Δ via the following formula:

$\begin{matrix} {T^{*} = {\underset{T}{argmin}{\sum\limits_{i = {1\ldots\; L}}{w\left( {v_{i},{S\left( v_{i} \right)}} \right)}}}} & {{Equa}.\mspace{14mu} 6} \\ {{{{subject}\mspace{14mu}{to}\mspace{14mu}{{{S\left( v_{i} \right)} - {S\left( v_{j} \right)}}}} \leq \Delta},} & \; \end{matrix}$ where Δ is a smoothness constraint that requires that the surface's position between neighboring vertices v_(i) and v_(j) does not change more than a constant distance Δ.

An optimal solution to the surface optimization formulation as defined in Equa. 6 can be transformed into a minimum-cost path formulation within a node-weighted graph, which was first solved in work by Wu, X., Chen, D. Z.: Optimal Net Surface Problems With Applications (see Proc. 29th Intl Colloquium Automata, Languages, and Programming (ICALP). p. 10291042 (2002)) (hereinafter referred to as Wu). In Wu, the minimum-cost path is sought by finding the most upper graph nodes of a minimum closed set. The Wu method is then utilized in solving a segmentation problem. In an embodiment disclosed herein, a similar graph representation as proposed in Wu is utilized. FIG. 5(d) depicts such a graph representation as employed in an embodiment herein. Each column of the graph of FIG. 5(d) represents a vertex (V1, V2, etc.) of the mesh 504 and corresponds to a searching profile. Graph nodes (N1, N2, etc.) represent sampling points. Cost w(v_(i), S(v_(i))) is assigned to each node. Via iteration of the method disclosed herein, the minimum closed set problem and its dual, the maximum flow problem, is solved by a polynomial-time implementation (algorithm), utilizing such methods as disclosed in Boykov, Y., Kolmogorov, V.: An Experimental Comparison Of Min-Cut/Max-Flow Algorithms For Energy Minimization In Vision (IEEE Trans Pattern Anal Mach Intell 26 (9), 1124-1137 (2004)).

With reference briefly back to FIGS. 5(a)-5(d), which provide an overview of graph structure reconstruction: FIG. 5(a) depicts a synthetic mesh surface (e.g. tubular shape) 502; FIG. 5(b) depicts a mesh zoom-in 504; FIG. 5(c) depicts vessel point sampling in 3-D space 506 as performed along a searching profile (e.g. vertex normal direction) for a vertex V_(i); and, FIG. 5 (d) depicts a graph structure 508 constructed based on sampling nodes. Graph nodes “N” of the same vertex “V” are intraconnected from top to bottom. The smoothness is enforced by inter-connecting nodes from neighboring columns under a distance A.

In the definition of Eq. 6, w(v_(i), S(v_(i))) is a cost function and describes the unlikeliness that a node is on the vessel surface. To resolve this cost function, we employ vessel boundary unlikeliness based cost w_(e)(v_(i), S(v_(i))) and vessel lumen (region) unlikeliness based cost w_(r)(v_(i), S(v_(i))). The combination of these two costs is formulated as follows: w(v _(i) ,S(v _(i)))=(α)(w _(e)(v _(i) ,S(v _(i))))+(1−α)(w _(r)(v _(i) ,S(v _(i)))),  Equa. 7 where α∈[0, 1] is a weighting (balance) parameter between boundary-based cost and region-based cost. The boundary-based cost is defined as follows: w _(e)(v _(i) ,S(v _(i)))=(1−p _(e)(v _(i) ,S(v _(i)))),  Equa. 8 where p_(e)∈[0, 1] is the probability that a node belongs to a vessel boundary. The probability p_(e) can be any normalized value depicting the likeliness of the vessel boundary. The region-based cost is defined as follows:

$\begin{matrix} {{{w_{r}\left( {v_{i},{S\left( v_{i} \right)}} \right)} = {{\sum\limits_{{S{(v_{i})}}^{\prime} \leq {S{(v_{i})}}}\left( {1 - {p_{v}\left( {v_{i},{S\left( v_{i} \right)}^{\prime}} \right)}} \right)} + {\sum\limits_{{S{(v_{i})}}^{\prime} > {S{(v_{i})}}}\left( {1 - {p_{nv}\left( {v_{i},{S\left( v_{i} \right)}^{\prime}} \right)}} \right)}}},} & {{Equa}.\mspace{14mu} 9} \end{matrix}$

where p_(v)∈[0, 1] is the probability that a node belongs to a vessel, and p_(nv)∈[0, 1] is the probability that a node does not belong to a vessel. The probability p_(v) can be any normalized value depicting the likeliness of the vessel, and the probability p_(v) can be any normalized value depicting the likeliness of the none vessel region.

In an embodiment, the probability p_(e) can be defined by the following equation:

$\begin{matrix} {{p_{e} = {\sum\limits_{j = 1}^{m}{u^{{S{({vi})}},j}{p_{e}^{j}\left( {Q\left( {a_{j},{S\left( v_{i} \right)}} \right)} \right)}}}},} & {{Equa}.\mspace{14mu} 10} \end{matrix}$ where a_(j) is computed based on Eq. 5, u^(S(vi),j) denotes the visibility of 3-D surface point S(vi) on image j, and p^(j) _(e) denotes a vessel boundary probability map of view j. In an embodiment, u^(S(vi),j) is a binary value of 1 or 0, where a value of 1 means the point u^(S(vi),j) is visible on the vessel boundary in image j, and a value of 0 means otherwise. In an embodiment, a soft visibility value is utilized as follows: u ^(S(vi),j)=exp((∠(n _(S(vi)) ,v _(j))−π/2)/σ_(u)),  Equa. 11 where ∠(n_(S(vi)), v_(j)) is the angle between the normal direction n_(S(vi)) at a predicted surface point S(v_(i)) and view direction v_(j) of camera j, and σ_(u) is a standard deviation. In an embodiment, p^(j) _(e) can be a probability map generated using a machine learning based vessel boundary detector. In an embodiment, p^(j) _(e) is a normalized gradient map, and in another embodiment, both gradient magnitude and direction are considered as follows:

$\begin{matrix} {{p_{e}^{j}\left( {Q\left( {a_{j},{S\left( v_{i} \right)}} \right)} \right)} = \begin{matrix} \left\{ {0,} \right. & {{{{if}\mspace{14mu}\left\lbrack {n_{Q{({{aj},{S{({vi})}}})}} \cdot {g_{dir}^{j}\left( {Q\left( {a_{j},{S\left( v_{i} \right)}} \right)} \right)}} \right\rbrack} < 0},} \\ {g_{mag}^{j}\left( {{{Q\left( {a_{j},{S\left( v_{i} \right)}} \right)}0};} \right.} & {\left. {{if}\mspace{14mu}{otherwise}} \right\},} \end{matrix}} & {{Equa}.\mspace{14mu} 12} \end{matrix}$ where n_(Q(aj, S0(vi))) is defined in Eq. 13 below, g^(j) _(dir)(Q(a_(j), S(v_(i)))) and g^(j) _(mag)(Q(a_(j), S(v_(i)))) are the gradient direction and gradient magnitude at 2-D location Q(a_(j), S(v_(i))) in image j, respectively, and the symbol “·” represents a dot product of the two associated vectors. n _(Q(aj, S0(vi))) =<Q(a _(j) ,S _(l)(v _(i))),Q(a _(j) ,S(v _(i)))>,  Equa. 13 where operator <, > in Equa. 13 represents a vector from the first associated parameter of Q( ) to the second associated parameter of Q( ), and S_(l)(v_(i)) is any point on searching profile of vertex v_(i) below S(v_(i)) (see FIGS. 5(a)-5(d) for example), where the noted term “below” herein means near to the vessel centerline.

In an embodiment, p_(v) is defined as follows:

$\begin{matrix} {{p_{v} = {\sum\limits_{j = 1}^{m}{w^{j}{p_{v}^{j}\left( {Q\left( {a_{j},{S\left( v_{i} \right)}} \right)} \right)}}}},} & {{Equa}.\mspace{14mu} 14} \end{matrix}$

where p^(j) _(v) denotes a vessel region probability map of view j, and w^(j) defines the weight of an individual projected response (p^(j) _(v)) in the vessel region confidence p_(v).

In an embodiment, p_(nv) is defined as follows:

$\begin{matrix} {{p_{nv} = {\sum\limits_{j = 1}^{m}{w^{j}{p_{n\; v}^{j}\left( {Q\left( {a_{j},{S\left( v_{i} \right)}} \right)} \right)}}}},} & {{Equa}.\mspace{14mu} 15} \end{matrix}$ where p^(j) _(nv) denotes a known vessel region probability map of view j, w^(j) defines the weight of an individual projected response (p^(j) _(nv)) in the known vessel region confidence p_(nv).

In an embodiment, p_(nv)=(1−p_(v)), thereby avoiding computation of Eq. 15. In an embodiment, p^(j) _(v) is a probability map generated using a machine learning based vessel detector. In an embodiment, p^(j) _(v) can be a normalized vesselness response map based on a method proposed in the work of Frangi, A. F., Niessen, W. J., Vincken, K. L., Viergever, M. A.: Multiscale Vessel Enhancement Filtering, in: MICCAI 1998, pp. 130-137 (1998).

The smoothness constraint when starting with the uniform surface defined by the sampling points of FIG. 4 may result in the target or reconstructed surface having less detail. A hierarchal approach in 3-D surface reconstruction adds a step to increase capture of surface detail. In the first iteration to fit the surface, the uniform sampled radius and the smoothness constraint is rather strict to get the initial vessel structure. To obtain a more realistic vessel surface with greater detail, an additional hierarchical step is applied with a less strict smoothness constraint and more flexibility adapting to the 2-D scene or projections and/or the vesselness images.

An adaptive radius is determined for each 3-D centerline point of the 3-D centerline created from the 2-D projections. The radii are estimated from the initial reconstructed 3-D surface. The 3-D surface points reconstructed in the first or a later iteration are clustered to their nearest 3-D centerline points. For each 3-D centerline point, the median of the radius from the centerline point to the points of the cluster is used as the sampling radius.

The 3-D sampling points are defined from the reconstructed centerlines or 3-D vessel skeleton tree. From 3-D centerlines reconstructed by the 2-D vessel points, a circular sample of points is formed around each of the 3-D centerline point, such as shown in FIG. 4. Unlike FIG. 4, different 3-D centerline points may have different radii. The radius as the median estimated from the clustering of the surface points for each centerline point is used. Rather than using the sampling points of FIG. 4 with the same radius applied to each centerline point, sampling points around different centerline points may have different radii. Using this variation in the sampling points allows for greater surface detail since the smoothness limitation is based on the sampling locations.

Given the sampling points defined based on the different radii by 3-D centerline location, the fitting continues as discussed above. A Poisson surface reconstruction method is utilized to compute the mesh from the point cloud. One or more additional iterations are performed to reconstruct the surface with Poisson reconstruction and matching. From the sampling points around the 3-D centerline with coordinates and geometric information (e.g., normal), a Poisson surface is constructed with newly formed faces. Then, the surface points are matched to the nearest sampling point to get the projected image and vesselness gradient as the response values. Other fitting using the sampling points may be used. Since the different radii found by clustering an initial surface reconstruction better match variance of the vessel surface, the resulting fit shows more surface detail. The limitations on deviation of the surface (i.e., smoothness limitations) of the cost function for fitting is from the partially fit surface points rather than a uniform cylinder.

The 2-D projections may include a catheter. The catheter may be opaque or visible in the 2-D projections, but is not itself part of the vessel tree. A penalization for the intersection from catheters may be added to the cost function in the Poisson method or other fitting. To avoid the diverging force from any background catheters intersecting with coronary vessels in 2-D projections, the angle formed by the surface normal and gradient angle at each centerline point is calculated. If the angle exceeds a threshold, (e.g., around π/2), the angle is identified as corrupted by the catheter intersection. When corrupted, an additional cost term may be added to penalize in reconstruction, such as a cost term based on the magnitude of the angle The vessel tree includes interconnected branches. The surface reconstruction may result in one or more disconnected branches from miss-identified outliers in the vessel segmentation. After the final surface is fit, any disconnected branch components may be eliminated. As an alternative to post-processing, any disconnected branches may be removed during the fitting or prior to fitting (e.g., remove from 3-D centerline).

From the face or outer surface information of the 3-D reconstructed surface, a breadth first search identifies connected parts of the 3-D surface. Other searching may be used, such as a connected component analysis. The largest connected component is found. For example, a threshold is applied to maintain a largest connected component. As another example, the size (e.g., volume and/or length) of the different connected components are compared. The smaller components or components below a threshold are eliminated as outliers. In another embodiment, small disconnected components are removed before surface reconstruction (e.g., removed at the centerline generation stage).

Experimental results of an embodiment of the method disclosed herein are depicted herein with reference to FIGS. 6(a)-6(l). These results do not use the affine or warping transforms and the variable radii for sampling points in the hierarchal reconstruction. FIGS. 6(a)-6(l) show examples of coronary vessel surface reconstruction from a limited number of 2-D rotational X-ray angiography images. For example, the first column, FIGS. 6(a), 6(d), 6(g) and 6(j), shows original images from four different patients. The second column, FIGS. 6(b), 6(e), 6(h) and 6(k), shows reconstructed vessel surfaces from the same view as acquired original images via an embodiment of the method disclosed herein. And, the third column, FIGS. 6(c), 6(f), 6(i) and 6(l), shows the projected 2-D segmentation overlays from reconstructed surfaces. As can be seen, the 3-D coronary artery tree surface reconstruction method disclosed herein using a limited number of 2-D rotational X-ray angiography images produces resulting images that correlate well with actual patient associated anatomies, as depicted by the third column of overlays in FIGS. 6(c), 6(f), 6(i) and 6(j).

With respect to the foregoing, it will be appreciated that a method to reconstruct a vessel surface from a limited number of rotational X-Ray angiography images is disclosed. The method disclosed compensates rigid and non-rigid transformations due to breathing and cardiac motion across different 2-D projection views. The method assumes that most of non-rigid vessel deformation is compensated for by ECG gating, that most of the rigid transformations are compensated for in a bundle adjustment based method, and that the remaining transformation differences are compensated for by an optimal graph search based optimization framework. An embodiment determines an optimal surface reconstruction in 3-D space, which avoids having to perform 2-D vessel segmentation and having to find 2-D vessel boundary correspondence.

Rather than assuming compensation for the rigid and non-rigid transformation, the registration of the centerline points from the 2-D projections with the CT 3-D centerline may use affine or thin-plate spline transformations. After segmenting the coronary vessel centerline points on 2D scenes, the point clouds are registered to each other with proper correspondence (see step 706 of FIG. 7). A 3-D centerline model from tomography construction (e.g., CT 3-D centerline) is used as reference between different 2-D frames or projections for determining correspondence between 2-D centerline points from different projections. A specific type of transformation is applied to the 3-D modelpoints before or after projection to a 2-D scene for matching. The transformation represents patient deformation or camera motion, such as heart contraction and breathing motion.

The Gaussian Mixture Model (GMM) is used for point cloud registration between the 3-D moving points and the 2-D scene centerline points, optimizing the fidelity of transformed and projected 3D model to the 2-D scene centerline points. Optimization aims at minimizing the L₂ or other cost between these two set of points. For example, the likelihood function may be minimized. Restating or formatting equation 2, the cost function becomes: F(X,Y,θ)=∫{gmm(X)−gmm[Π(Y _(θ)]}² dx  Equa. 16 where X is the point clouds for centerlines in the 2-D scenes or projections, Y is the moving point cloud (i.e., CT 3-D centerline) for registration in 3-D, θ represents the parameters for transformation (e.g., rotation R, translation T, scale A, and/or warp) and H is the perspective projection from 3-D to 2-D for each projection angle.

The gradient of a perspective projection H of point cloud Q in 3D is:

$\begin{matrix} {\frac{\partial{\prod(Q)}}{\partial Q} = {\frac{1}{z_{p}}\left\{ {\prod\limits_{{1:2},{1:3}}{- \frac{\prod\limits_{{1:2},{1:4}}{\left\lbrack Q \middle| 1 \right\rbrack^{T}\prod\limits_{3}}}{z_{p}}}} \right\}}} & {{Equa}.\mspace{14mu} 17} \end{matrix}$ where z_(p)=[Π(Q)]_(z). The transformation is used in this framework to minimize the cost function, registering the 2-D centerlines with the CT 3-D centerline for establishing correspondence of the centerline points of the different 2-D projections.

In one embodiment, a 2-D to 3-D rigid transformation is used. The 2-D to 3-D rigid transformation applies the transform to the CT 3-D centerline prior to projection. Under the assumption of rigid trans formation, rotation with an orthogonal matrix followed by translation is applied on the 3-D model points, then, projection with camera projection matrix is realized. This approach is represented as: Z=Π(R·Y+T)  Equa. 18 where R∈SO(3), and T∈

³. The rotation R and translation T have three degrees of freedom. Scale or warping are not included.

In other embodiments, affine transformations are used. For example, a 2-D to 2-D affine transformation is used. The 2-D to 2-D affine transformation applies the transform after projections of the CT 3-D centerline. 3-D model points are projected to each 2-D scene. Linear transformation with scaling and translation with or without rotation is applied in 2-D to the projected points form the CT 3-D. These transformed points are registered to the 2-D centerline points from the 2-D projections. This 2-D to 2-D affine transformation is represented as: Z=A·Π(Y)+T  Equa. 19 where A∈

^(2×2) and T∈

². A is the scale with or without rotation. Warping is not included.

In another example, a 2-D to 3-D affine transformation is used. The 2-D to 3-D affine transformation applies the transform to the CT 3-D centerline prior to projection. In comparison to the 2-D to 2-D affine transformation, linear transformation including 3-D scaling and translation with or without rotation is applied on the moving model points in 3-D, and projection to 2-D scene is then registered. This 2-D to 3-D affine transformation is represented as: Z=Π(A·Y+T)  Equa. 20 where A∈

^(3×3) and T∈

³. Warping is not included.

In another embodiment, a thin-plate spline (TPS) or other warping transformation is applied. In addition or as an alternative to adjusting scale in an affine transformation or just rotation and translation in a rigid transformation, the relative position of points to each other may be adjusted in a non-uniform manner by warping. While a 2-D to 2-D approach may be used in warping, one embodiment uses a 2-D to 3-D approach. In addition to the L₂ norm or other difference between the two point clouds used in the GMM, a regularization term with the internal structure of the points is added to reinforce the smoothness of the structure formed by transformed points. The amount or extent of warping is limited. Let Q=(q1, q2, . . . , qc)^(T) be the matrix representing a set of c control or centerline points of the CT 3-D centerline. The kernel matrix K={K_(ij)}={K(|q_(i)−q_(j)|)} describes the internal structure of the control points. Uniform spacing in 3-D for control points is more stable than using vessels from experiments. The TPS transform can be decomposed into a linear part with an affine motion A and a nonlinear part with TPS warping coefficients w, as below: M=[1|Y]A ^(T) +Uw=[1|Y]A ^(T) +UNv  Equa. 21 where the basis matrix is U={U_(ij)}={K(x_(i)−q_(j))}, and to ensure the boundary condition w^(T) [1|Q]=0, introduce N as the left null space of [1|Q] and w=Nv.

The gradient of the cost function F(M(θ)) with respect to the parameters θ can be computed with the chain rule:

$\begin{matrix} {\frac{\partial F}{\partial\theta} = {\frac{\partial F}{\partial{\prod(M)}} \cdot \frac{\partial{\prod(M)}}{\partial M} \cdot \frac{\partial M}{\partial\theta}}} & {{Equa}.\mspace{14mu} 22} \end{matrix}$ where

$G = {\frac{\partial F}{\partial M} \cdot \frac{\partial{\prod(M)}}{\partial M} \cdot \frac{\partial F}{\partial M}}$ can be obtained as the byproduct when evaluating the cost function F, and

$\frac{\partial{\prod(M)}}{\partial M}$ can be evaluated as the previously derived gradients for perspective projections.

The bending energy of the nonlinear transformation is trace(w^(T) Kw). To regularize the TPS nonrigid transform, a weighted penalty term is added to the final cost function and the gradients for parameters are thus obtained as follows:

$\begin{matrix} {\frac{\partial F}{\partial A} = {\left\lbrack 1 \middle| Y \right\rbrack^{T}G}} & {{Equa}.\mspace{14mu} 23} \\ {\frac{\partial F}{\partial v} = {{({UN})^{T}G} + {\lambda\; N^{T}{KNv}}}} & \; \end{matrix}$ where λ is the parameter adjusting the level of warping. Other affine and/or warping transformations may be used.

Using the rigid, affine, or warping transform for registering the CT 3-D centerline to the centerlines of the 2-D projections and the hierarchal added step of sampling based on clustering from an initial surface reconstruction in surface reconstruction, the surface reconstruction is evaluated. For example, the results from 22 dynaCT scans are qualitatively evaluated. In average, 5 to 9 ECG synchronized frames are used for each dynaCT scan. Optimization with gradients reduce the computational time by 20% as compared to without. The image processor may more quickly reconstruct the surface.

For the registration of the CT 3-D centerline (i.e., 3-D model from dynaCT) with the 2-D centerlines from the 2-D projections, the GMM is run with the different transformations on the 22 datasets. As point clouds representing the 3-D model are matched to segmented point clouds in 2-D representing the vessels in projections, the percentage of successfully registered points is evaluated as a measure of accuracy.

Transformation Type Mean Matched Ratio Standard Deviation Rigid 0.8129 0.1467 Affine in 2-D 0.8224 0.1456 Affine in 3-D 0.8459 0.1299 The affine transformation in 3-D (i.e., 2-D to 3-D affine transformation) performs the best. The TPS transformation is less stable.

For vessel surface reconstruction evaluation, the 2-D to 3-D affine transformation is used for the registration to create the centerline from the 2-D projections. The 3-D vessel surface is obtained after an initial reconstruction and a following hierarchical reconstruction based on sampling circles with radii based on medians of clustered initial surface points to centerline points followed by Poisson surface reconstruction. The vessel surface in 3-D can be reprojected to input images to observe reconstruction quality. FIGS. 9(a), 9(c), and 9(e) show examples of final reconstructed 3-D vessel surfaces, and FIGS. 9(b), 9(d), and 9(f) show example reprojections of the reconstructed 3-D vessel surfaces onto the x-ray or projection images. Due to the sampling points based on different radii, bifurcations are less bulky, resembling the underlying vessel geometry. Better separation of the vessel surfaces is provided as compared to reconstruction with just one radius applied to all the centerline points. The surface detail or variation in the surface is greater by using the sampling points based on different radii. Sampling points generated according to a previously reconstructed surface allow more flexibility in the following graph search step and thus establish a surface with more details included.

The greater surface detail may provide valuable information for physicians deciding whether or where to place a stent. Stenosis is a sensitive region to compute fractional flow reserve, so the greater detail leads to more accurate non-invasive calculation of fractional flow reserve. Rotational angiographic x-ray images may be used to determine fractional flow reserve or other quantifications for stenosis or the vessel tree due to the ability to more accurately segment or reconstruct the 3-D vessel surface.

In view of the foregoing, and with reference to FIG. 7, it will be appreciated that an embodiment of the invention includes the following. An embodiment includes a method 700 for reconstructing 3-D vessel geometry of a vessel, comprising: receiving 702 a plurality of 2-D rotational X-ray images of the vessel; extracting 704 vessel centerline points for normal cross sections of each of the plurality of 2-D images; establishing 706 a correspondence of the centerline points from a registration of the centerline points with a computed tomography (CT) 3-D centerline, the registration being an affine or deformable transformation; constructing 708 a 3-D centerline vessel tree skeleton of the vessel from the centerline points of the 2-D images; constructing 710 an initial 3-D vessel surface from a uniform radius normal to the 3-D centerline vessel tree skeleton; defining 711 sample points based sampling on median radii to the 3-D centerline vessel tree skeleton of the initial 3-D vessel surface and constructing 712 a target 3-D vessel surface by deforming the initial vessel surface using the sample points to provide a reconstructed 3-D vessel geometry of the vessel. An additional act to remove 713 disconnected branch components of the target 3-D vessel surface may be provided.

An embodiment includes a computer program storage medium 720 comprising a non-transitory computer readable medium having program code executable by a processing circuit for implementing the method 700.

An embodiment includes an apparatus 750 for reconstructing 3-D vessel geometry of a vessel, comprising: a computer controlled machine 752 comprising a processing circuit 754; computer readable executable instructions 720, which upon loading into the processing circuit causes the machine to be responsive to the executable instructions to facilitate implementation of the method 700. The apparatus 750 may be an angiography imaging system.

In an embodiment, the method 700 is facilitated by the processing circuit 754 (e.g., image processor) that is responsive to machine readable executable instructions 720 which when executed by the processing circuit 754 facilitates at least the steps 702, 704, 706, 708, 710, 712 of the method 700 to reconstruct 3-D vessel geometry of the vessel 106.

In view of the foregoing, and with reference to FIGS. 8(a), 8(b) and 8(c), it will be appreciated that an apparatus 800 is used to capture 2-D rotational X-ray angiography images 802 of a patient 806 in accordance with an embodiment using a 2-D rotational X-ray machine (also herein referred to as a camera) 804, where the 2-D rotational X-ray machine 804 is controlled using ECG signals or ECG gated acquisition signals 808 from ECG machine 810. FIG. 8(a) depicts a side view the patient 806 and camera 804, FIG. 8(b) depicts an end view of the same patient 806 and camera 804 shown in four positions, and FIG. 8(c) depicts images 802 produced by the camera 804. A first set of images 802.1 are received from a first camera position 804.1, a second set of images 802.2 are received from a second camera position 804.2, and an nth set of images 802.n are received from an nth camera position 804.n. Ellipses 812 represent a plurality of respective sets of images 802 from a given camera 804 position, and ellipses 814 represent a plurality of respective sets of images 802 from a plurality of positions of a given camera 804 as it rotates 816 about the patient 806.

In view of the foregoing, it will be appreciated that an embodiment may be embodied in the form of computer-implemented processes and apparatuses for practicing those processes. An embodiment may also be embodied in the form of a computer program product having computer program code containing instructions embodied in tangible media, such as floppy diskettes, CD-ROMs, hard drives, USB (universal serial bus) drives, or any other computer readable storage medium, such as random access memory (RAM), read only memory (ROM), erasable programmable read only memory (EPROM), electrically erasable programmable read only memory (EEPROM), or flash memory, for example, wherein, when the computer program code is loaded into and executed by a computer, the computer becomes an apparatus for practicing the invention. An embodiment may also be embodied in the form of computer program code, for example, whether stored in a storage medium, loaded into and/or executed by a computer, or transmitted over some transmission medium, such as over electrical wiring or cabling, through fiber optics, or via electromagnetic radiation, wherein when the computer program code is loaded into and executed by a computer, the computer becomes an apparatus for practicing the invention. When implemented on a general-purpose microprocessor, the computer program code segments configure the microprocessor to create specific logic circuits. A technical effect of the executable instructions is to reconstruct a 3-D coronary artery tree surface from a limited number of 2-D rotational X-ray angiography images.

While the invention has been described with reference to exemplary embodiments, it will be understood by those skilled in the art that various changes may be made and equivalents may be substituted for elements thereof without departing from the scope of the claims. In addition, many modifications may be made to adapt a particular situation or material to the teachings of the invention without departing from the essential scope thereof. Therefore, it is intended that the invention not be limited to the particular embodiment disclosed as the best or only mode contemplated for carrying out this invention, but that the invention will include all embodiments falling within the scope of the appended claims. Also, in the drawings and the description, there have been disclosed example embodiments and, although specific terms may have been employed, they are unless otherwise stated used in a generic and descriptive sense only and not for purposes of limitation, the scope of the claims therefore not being so limited. Moreover, the use of the terms first, second, etc. do not denote any order or importance, but rather the terms first, second, etc. are used to distinguish one element from another. Furthermore, the use of the terms a, an, etc. do not denote a limitation of quantity, but rather denote the presence of at least one of the referenced item. Additionally, the term “comprising” as used herein does not exclude the possible inclusion of one or more additional features. 

What is claimed is:
 1. A method for reconstructing 3-D vessel geometry of a vessel, the method comprising: receiving a plurality of 2-D rotational X-ray images of the vessel; extracting vessel centerline points for normal cross sections of each of the plurality of 2-D images; establishing a correspondence of the centerline points from a registration of the centerline points with a computed tomography (CT) 3-D centerline, the registration being an affine or deformable transformation; constructing a 3-D centerline vessel tree skeleton of the vessel from the centerline points of the 2-D images; constructing an initial 3-D vessel surface from a uniform radius normal to the 3-D centerline vessel tree skeleton; defining sample points based sampling on median radii to the 3-D centerline vessel tree skeleton of the initial 3-D vessel surface; and constructing a target 3-D vessel surface by deforming the initial vessel surface using the sample points to provide a reconstructed 3-D vessel geometry of the vessel.
 2. The method of claim 1, wherein the vessel is a coronary artery and the 2-D images have a same cardiac phase.
 3. The method of claim 1, wherein the 2-D images are angiography images.
 4. The method of claim 1, wherein: the extracting vessel centerlines is accomplished in a form of ordered point sets, S_(i), from angiographic images, where i is the i^(th) image, and each point set S_(i) is a projection of a transformed 3-D centerline model M according to the following: S _(i) =P _(i) T(M,θ _(i)), wherein P_(i) is a camera projection operator corresponding to image i, and T(M, θ_(i)) is a defined transformation model; wherein given M and S_(i), Gaussian mixture models (GMM) are employed to find transformations T(M, θ_(i)) that minimize the following cost function: J(S _(i) ,M,θ _(i))=∫(GMM(S _(i))−GMM(P _(i) T(M,θ _(i))))^(p) dx, wherein J(Si,M, θ_(i)) is an energy function that measures a distance of two Gaussian mixture models (GMM) where p∈[0, 2].
 5. The method of claim 4, wherein: the establishing the correspondence of centerline points is accomplished using a bipartite graph that corresponds to a linear assignment process that finds a matching vessel point for every projected model point in each image, wherein the bipartite graph represents a first set of vertices (i) that represent transformed, projected model points of the vessel centerline points, and a second set of vertices (j) that represent associated segment points of the CT 3-D centerline from a CT volume and CT reconstruction, wherein weights at edges, cij, of paired vertices of the first and second sets of vertices are computed using a distance of spatial locations of the respective vertices d_(ij) according to the following: c _(i,j) ={d _(i,j) if d _(i,j) <dth; ∞ if d _(i,j—) d _(th)}, wherein d_(th) is a defined distance threshold.
 6. The method of claim 5, wherein: the constructing the 3-D centerline vessel tree skeleton is accomplished according to the following: ${\min\limits_{a_{j},b_{i}}{\sum\limits_{i = 1}^{n}{\sum\limits_{j = 1}^{m}{k^{i,j}{d\left( {{Q\left( {a_{j},b_{i}} \right)},x^{i,j}} \right)}}}}},$ wherein: n is a number of 3-D points that are seen in m views; camera (j) is parameterized by a vector a_(j); b_(i) is the i^(th) centerline point; Q(a_(j), b_(i)) is the projection of 3-D point b_(i) on image (j); x^(i,j) is the i^(th) 2-D centerline point measurement on image (j); d(,) is a Euclidean distance between the associated points Q(a_(j), b_(i)) and x^(i,j); and, k^(i,j)∈[0, 1] is a confidence measurement of x^(i,j).
 7. The method of claim 6, wherein: the constructing the initial 3-D vessel surface is accomplished by generating a point cloud of circles around and perpendicular to a tangential direction of the associated centerline, each one of the point cloud of circles having a defined radius r, and computing an initial surface mesh via a Poisson surface reconstruction method using the point cloud of circles.
 8. The method of claim 1, wherein: the extracting vessel centerlines is accomplished using a technique of model-guided extraction.
 9. The method of claim 1, wherein: the establishing the correspondence of centerline points is accomplished using the affine transformation, the affine transformation comprising a 2-D to 2-D affine transformation or a 2-D to 3-D affine transformation.
 10. The method of claim 1, wherein: the establishing the correspondence of centerline points is accomplished using the thin-plate spline transformation.
 11. The method of claim 1, wherein: the constructing the 3-D centerline vessel tree skeleton is accomplished using 2-D points of the extracted vessel centerlines using a bundle adjustment based approach.
 12. The method of claim 1, wherein: the defining the sample points is accomplished by clustering points of the initial surface mesh to nearest points of the 3-D centerline vessel tree skeleton, calculating the median radius for each of the points of the 3-D centerline vessel tree skeleton to the points of the initial surface mesh clustered to the point of the 3-D centerline vessel tree skeleton, and defining the sample points for each of the points of the 3-D centerline vessel tree skeleton as along a circle with the median radius for the respected 3-D centerline vessel tree skeleton.
 13. The method of claim 12, wherein: the constructing the target 3-D vessel surface is accomplished using 2-D information through 3-D to 2-D projection to deform the sample points according to the following: T={S(v _(i))|i=1 . . . L} wherein T represents a surface of the sample points, where L is the number of vertices (v_(i)), i is an iterative parameter, and S(v_(i)) is a sampling point along a searching profile for (v_(i)), wherein a minimum-surface cost function w(v_(i), S(v_(i))), subject to smoothness constraints |S(v_(i))−S(v_(i))|≤Δ, is solved via the following: $\begin{matrix} {T^{*} = {\underset{T}{argmin}{\sum\limits_{i = {1\ldots\; L}}{w\left( {v_{i},{S\left( v_{i} \right)}} \right)}}}} & \; \\ {{{{subject}\mspace{14mu}{to}\mspace{14mu}{{{S\left( v_{i} \right)} - {S\left( v_{j} \right)}}}} \leq \Delta},} & \; \end{matrix}$ wherein T* represents the target 3-D vessel surface, and A is a smoothness constraint that requires that the surface's position between neighboring vertices (v_(i)) and (v_(j)) does not change more than a constant distance Δ.
 14. The method of claim 13, wherein the cost function w(v_(i), S(v_(i))) describes the unlikeliness that a node is on the target vessel surface and is resolved according to the following: w(v _(i) ,S(v _(i)))=(α)(w _(e)(v _(i) ,S(v _(i))))+(1−α)(w _(r)(v _(i) ,S(v _(i)))), wherein α∈[0, 1] is a weighting (balance) parameter between a boundary-based cost w_(e)(v_(i), S(v_(i))) and region-based cost w_(r)(v_(i), S(v_(i))), the boundary-based cost being defined as follows: w _(e)(v _(i) ,S(v _(i)))=(1−p _(e)(v _(i) ,S(v _(i)))), wherein p_(e)∈[0, 1] is the probability that a node belongs to a vessel boundary, wherein the probability p_(e) is a normalized value depicting the likeliness of the vessel boundary, wherein the region-based cost is defined as follows: ${{w_{r}\left( {v_{i},{S\left( v_{i} \right)}} \right)} = {{\sum\limits_{{S{(v_{i})}}^{\prime} \leq {S{(v_{i})}}}\left( {1 - {p_{v}\left( {v_{i},{S\left( v_{i} \right)}^{\prime}} \right)}} \right)} + {\sum\limits_{{S{(v_{i})}}^{\prime} > {S{(v_{i})}}}\left( {1 - {p_{nv}\left( {v_{i},{S\left( v_{i} \right)}^{\prime}} \right)}} \right)}}},$ wherein p_(v)∈[0, 1] is the probability that a node belongs to a vessel, and p_(nv)∈[0, 1] is the probability that a node does not belong to a vessel.
 15. The method of claim 14, wherein: probability p_(e) is defined by the following: $\begin{matrix} {{p_{e} = {\sum\limits_{j = 1}^{m}{u^{{S{({vi})}},j}{p_{e}^{j}\left( {Q\left( {a_{j},{S\left( v_{i} \right)}} \right)} \right)}}}},} & \; \end{matrix}$ wherein u^(S(vi),j) denotes a visibility of a 3-D surface point S(vi) on image j, and p^(j) _(e) denotes a vessel boundary probability map of view j, wherein u^(S(vi),j) is a binary value of 1 or 0, where a value of 1 means the point u^(S(vi),j) is visible on the vessel boundary in image j, and a value of 0 means otherwise; wherein a soft visibility value is utilized as follows: u ^(S(vi),j)=exp((∠(n _(S(vi)) ,v _(j))−π/2)/σ_(u)), wherein ∠(n_(S(vi)), v_(j)) is an angle between a normal direction n_(S(vi)) at a predicted surface point S(v_(i)) and view direction v_(j) of camera j, and σ_(u) is a standard deviation; wherein p^(j) _(e) is either: a probability map generated using a machine learning based vessel boundary detector; a normalized gradient map; or, both gradient magnitude and direction are considered as follows: ${{p_{e}^{j}\left( {Q\left( {a_{j},{S\left( v_{i} \right)}} \right)} \right)} = \begin{Bmatrix} {0,} & {{{{if}\mspace{14mu}\left\lbrack {n_{Q({{aj},{S{({vi})}}}} \cdot {g_{dir}^{j}\left( {Q\left( {a_{j},{S\left( v_{i} \right)}} \right)} \right)}} \right\rbrack} < 0},} \\ {{g_{mag}^{j}\left( {Q\left( {a_{j},{S\left( v_{i} \right)}} \right)} \right)};} & {{if}\mspace{14mu}{otherwise}} \end{Bmatrix}},$ wherein g^(j) _(dir)(Q(a_(j), S(v_(i)))) and g^(j) _(mag)(Q(a_(j), S(v_(i)))) are a gradient direction and gradient magnitude at 2-D location Q(a_(j), S(v_(i))) in image j, respectively, wherein n_(Q(aj, S0(vi))) is defined as follows: n _(Q(aj, S0(vi))) =<Q(a _(j) ,S _(l)(v _(i))),Q(a _(j) ,S(v _(i)))>, wherein S_(l)(v_(i)) is any point on searching profile of vertex v_(i) below S(v_(i)).
 16. A computer program storage medium comprising a non-transitory computer readable medium having program code executable by a processing circuit for implementing the method according to claim
 14. 17. An apparatus for reconstructing 3-D vessel geometry of a vessel, comprising: a computer controlled machine comprising a processing circuit; computer readable executable instructions, which upon loading into the processing circuit causes the machine to be responsive to the executable instructions to facilitate implementation of the method according to claim
 14. 18. The method of claim 1, further comprising: penalizing in the constructing of the target 3-D vessel surface based on an angle between a surface normal and a gradient angle at each point of the 3-D centerline vessel skeleton tree.
 19. The method of claim 1, further comprising: removing disconnected branch components of the target 3-D vessel surface.
 20. The method of claim 1, wherein: the constructing the target 3-D vessel surface is accomplished using a polynomial-time implementation. 